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In this paper we study the Finsler sphere $(S^n,F)$ with $n>1$, which hasconstant flag curvature $K\equiv 1$ and only finite prime closed geodesics. Inthis case, the connected isometry group $I_0(S^n,F)$ must be a torus whichdimension satisfies $0<\dim I(S^n,F) \leq[\frac{n+1}{2}]$. We will prove thatthe number of geometrically distinct reversible closed geodesics on $(S^n,F)$is at least $\dim I(S^n,F)$. When $\dim I_0(S^n,F)=[\frac{n+1}{2}]$, theequality happens, and there are exactly $2[\frac{n+1}{2}]$ prime closedgeodesics, which verifies Anosov conjecture in this special case.

The number of geometrically distinct reversible closed geodesics on a Finsler sphere with $K\equiv 1$